new high-resolution central schemes for nonlinear conservation laws and convection diffusion...

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Journal of Computational Physics 160, 241282 (2000)

doi:10.1006/jcph.2000.6459, available online at http://www.idealibrary.com on

New High-Resolution Central Schemesfor Nonlinear Conservation Laws and

ConvectionDiffusion Equations

Alexander Kurganov and Eitan TadmorDepartment of Mathematics, University of Michigan, Ann Arbor, Michigan 48109;

andDepartment of Mathematics, UCLA, Los Angeles, California 90095

E-mail: [email protected],[email protected]

Received April 8, 1999; revised December 8, 1999

Central schemes may serve as universal finite-difference methods for solving non-

linear convectiondiffusion equations in the sense that they are not tied to the specificeigenstructure of the problem, and hence can be implemented in a straightforward

manner as black-box solvers for general conservation laws and related equations gov-

erning the spontaneous evolution of large gradient phenomena. The first-order Lax

Friedrichs scheme (P. D. Lax, 1954) is the forerunner for such central schemes. The

central NessyahuTadmor (NT) scheme (H. Nessyahu and E. Tadmor, 1990) offers

higher resolution while retaining the simplicity of the Riemann-solver-free approach.

The numerical viscosity present in these central schemes is of order O((x)2r/t).

In the convective regime where tx , the improved resolution of the NT schemeand its generalizations is achieved by lowering the amount of numerical viscositywith increasing r. At the same time, this family of central schemes suffers from

excessive numerical viscosity when a sufficiently small time step is enforced, e.g.,

due to the presence of degenerate diffusion terms.

In this paper we introduce a new family of central schemes which retain the sim-

plicity of being independent of the eigenstructure of the problem, yet which enjoy

a much smaller numerical viscosity (of the corresponding order O(x)2r1)). Inparticular, our new central schemes maintain their high-resolution independent of

O(1/t), and letting t 0, they admit a particularly simple semi-discrete formu-lation. The main idea behind the construction of these central schemes is the use of

more precise information of the local propagation speeds. Beyond these CFL related

speeds, no characteristic information is required. As a second ingredient in their

construction, these central schemes realize the (nonsmooth part of the) approximate

solution in terms of its cell averages integrated over the Riemann fans of varying

size.

The semi-discrete central scheme is then extended to multidimensional problems,

with or without degenerate diffusive terms. Fully discrete versions are obtained with

241

0021-9991/00 $35.00Copyright c 2000 by Academic Press

All rights of reproduction in any form reserved.


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242 KURGANOV AND TADMOR

RungeKutta solvers. We prove that a scalar version of our high-resolution central

scheme is nonoscillatory in the sense of satisfying the total-variation diminishing

property in the one-dimensional case and the maximum principle in two-space di-

mensions. We conclude with a series of numerical examples, considering convex

and nonconvex problems with and without degenerate diffusion, and scalar and sys-

tems of equations in one- and two-space dimensions. Time evolution is carried out

by the third- and fourth-order explicit embedded integration RungeKutta methods

recently proposed by A. Medovikov (1998). These numerical studies demonstrate

the remarkable resolution of our new family of central scheme. c 2000 Academic PressKey Words: hyperbolic conservation laws; multidimensional systems; degenerate

diffusion; central difference schemes; non-oscillatory time differencing.

CONTENTS

1. Introduction.

2. Central schemesA brief overview.

3. The fully discrete schemeOne-dimensional setup.

4. The reduction to semi-discrete formulation. 4.1. One-dimensional hyperbolic conservation laws. 4.2. One-

dimensional convection-diffusion equations. 4.3. Multidimensional extensions.

5. From semi-discrete back to fully discreteThe general setup.

6. Numerical examples. 6.1. One-dimensional scalar linear hyperbolic equation. 6.2. One-dimensional

scalar hyperbolic conservation laws. 6.3. One-dimensional systems of hyperbolic conservation laws.

6.4. One-dimensional convectiondiffusion equations. 6.5. Two-dimensional problems.

1. INTRODUCTION

During the past few decades there has been an enormous amount of activity related to the

construction of approximate solutions for nonlinear conservation laws,

tu(x , t)+

xf(u(x, t)) = 0, (1.1)

and for the closely related convectiondiffusion equations,

tu(x , t)+

xf(u(x, t)) =

xQ[u(x, t), ux (x, t)]. (1.2)

Here, u(x , t)= (u1(x, t) , . . . , uN(x , t)) is an N-vector of conserved quantities, f(u) is anonlinear convection flux, and Q(u, ux ) is a dissipation flux satisfying the (weak) parabol-

icity conditions Q(u, s) 0 u, s. In the general multidimensional case u isan N-vector inthe d-spatial variablesx = (x1, . . . ,xd),withthecorrespondingfluxes f(u)= ( f1, . . . , fd)and Q(u,x u)= (Q1, . . . , Qd).

These equations are of great practical importance since they govern a variety of physical

phenomena that appear in fluid mechanics, astrophysics, groundwater flow, meterology,

semiconductors, and reactive flows. Convectiondiffusion equations (1.2) also arise in two-phase flow in oil reservoirs, non-Newtonian flows, front propagation, traffic flow, financial

modeling, and several other areas.

In this work we present new second-order central difference approximations to (1.1)

and (1.2). These new schemes can be viewed as modifications of the NessyahuTadmor

(NT) scheme [38]. Our schemes enjoy the major advantages of the central schemes over


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HIGH-RESOLUTION CENTRAL SCHEMES 243

the upwind ones: first, no Riemann solvers are involved, and secondas a result of being

Riemann solver freetheir realization and generalization for complicated multidimensional

systems (1.1) and (1.2) are considerably simpler than in the upwind case. At the same time,

the new schemes have a smaller amount of numerical viscosity than the original NT scheme,

and unlike other central schemes, they can be written and efficiently integrated in their semi-

discrete form.

We would like to emphasize the importance of semi-discrete formulations for solv-

ing real, practical problems associated with multidimensional systems (1.1) and (1.2).

Semi-discrete schemes are especially effective when they combine high-resolution, non-

oscillatory spatial discretization with high-order, large stepsize ODE solvers for their time

evolution.

The advantage of our semi-discrete scheme is clearly demonstrated later, in Figs. 6.21 and6.22, where it is compared with the fully discrete NT solution of the degenerate convection

diffusion equation.

ut+ f(u)x =

ux1+ u2x

x

. (1.3)

This model, recently proposed in [28], describes high-gradient phenomena with possible

discontinuous subshock solutions; consult [12, 28] for details. When the NT scheme is used

to resolve these discontinuities, the computed subshocks are smeared due to the larger nu-

merical dissipation which is accumulated ove the small time steps enforced by the restricted

CFL stability condition, t (x )2. This situationof excessive numerical dissipation (oforder O((x)2r/t))is typical for fully discrete central schemes with time steps much

smaller than the convective CFL limitation. Alternatively, our new central scheme will ac-

cumulate less dissipation (of order O(x )2r1) and hence can be efficiently used with time

steps as small as required.This paper is organized as follows. In Section 2 we provide a brief description of the

central differencing approach for hyperbolic conservation laws.

In Section 3 we introduce our new fully discrete second-order central scheme, which

is constructed for systems of one-dimensional hyperbolic conservation laws. The limiting

case, t 0, brings us to the semi-discrete version presented in Subsection 4.1. Here weprove that our second-order semi-discrete central scheme satisfies the scalar total-variation

diminishing (TVD) property; consult Theorem 4.1 below. In Subsections 4.2 and 4.3 our

semi-discrete scheme is extended, respectively, to one-dimensional convectiondiffusion

equations and to multidimensional hyperbolic and (degenerate) parabolic problems.

In Section 5 we return to the fully discrete framework, discussing time discretization

for our semi-discrete central scheme. Specifically, we use efficient RungeKutta ODE

solvers to integrate the semi-discrete schemes outlined earlier in Section 4. We retain the

overall Riemann-free simplicity without giving up high resolution. Here we prove that

the resulting second-order fully discrete central scheme satisfies the scalar the maximum

principle; consult Theorem 5.1.We end in Section 6 by presenting a number of numerical results. These results are

convincing illustrations that our new central schemes provide high resolution at a low-

est cost, when applied both to hyperbolic systems of conservation laws and to a variety

of convectiondiffusion models. These numerical results confirm an essential aspect of

the current approachretaining high-resolution withoutthe costly (approximate) Riemann


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solvers, characteritic decompositions, etc. This aspect in the context of high-resolution

schemes was introduced in the NessyahuTadmor scheme [38] and was extended to two-

dimensional problems in [18]. A nonstaggered and hence less dissipative version was pre-

sented in [17]. The relaxation scheme introduced in [19] is closely related to these staggered

central schemes; in fact, they coicide in the relaxation limit 0. The choice for a relax-ation matrix A in [19] provides us a family of high-resolution schemes; we note in passing

that the special scalar choice A = (f(u)/u)I is an O() perturbation of the centralscheme discussed in this paper. Other componentwise approaches were presented in [34].

The CUSP scheme presented in [16] is a semi-discrete scheme which avoids characteristic

decompositions. And more recently, Liu and Osher [35] introduced a semi-discrete scheme

based on a pointwise formulation of ENO which retains high resolution without Riemann

solvers.

2. CENTRAL SCHEMESA BRIEF OVERVIEW

Central schemes offer universal finite-difference methods for solving hyperbolic con-

servation laws, in the sense that they are not tied up to the specific eigenstructure of the

problem and hence can be implemented in a straightforward manner as a black-box solver

for general systems (1.1). In particular, they do not involve characteristic decomposition of

the flux f. In fact, even computation of the Jacobian of f can be avoided; in the particular

case of the second-order NT scheme, for example, numerical derivatives of the flux in (2.6)

below can be implemented componentwiseconsult [18, 36].

In 1954 Lax and Friedrichs [10, 29] introduced the first-order stable central scheme, the

celebrated LaxFriedrichs (LxF) scheme:

un+1j=

unj+1 + unj12

2funj+1

funj1. (2.1)Here, :=t/x is the fixed mesh ratio, and unj is an approximate value ofu(x = xj , t=tn ) atthegridpoint(xj := j x, tn := nt). Compared with the canonical first-order upwindscheme of Godunov [11], the central LxF scheme has the advantage of simplicity, since

no (approximate) Riemann solvers, e.g., [42], are involved in its construction. The main

disadvantage of the LxF scheme, however, lies in its large numerical dissipation, which

prevents sharp resolution of shock discontinuities and rarefaction tips.

A natural high-order extension of the LxF schemethe NT schemewas presented in

1990 in [38]. The main idea of this generalization is replacing the first-order piecewise

constant solution which is behind the original LxF scheme with van Leers MUSCL-type

piecewise-linear second-order approximation, e.g., [31]. This is then combined with a LxF

solveran alternative to the upwind solvers, which avoids the time-consuming resolution

of Riemann fans by staggered (x, t)-integration. Thus, the NT scheme retains the simplicity

of the Riemann-free LxF framework while gaining high resolution, which eliminates the

disadvantage of excessive first-order dissipation.

Here is a brief readers digest based on the representation of the LxF and NT schemesas Godunov-type schemes. We follow [36, Section 2]. To this end, we utilize the sliding

average ofu(, t),

u(x, t) := 1|Ix |Ix

u(, t) d , Ix :=

: | x| x2

,


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so that the integration of (1.1) over the rectangle Ix [t, t+ t] yields an equivalentreformulation of the conservation law, (1.1),

u(x, t+t)= u(x , t)1

x t+t

=tf

u

x +x

2 ,

dt+t

=tf

u

xx

2 ,

d

.

(2.2)

We begin by assuming that we have already computed an approximation to the solution at

time level t= tn a piecewise linear approximation u(x, tn) u(x, tn) of the form

u(x , t

n

) :=j un

j + (ux )n

j (x xj )1[xj1/2,xj+1/2], xj 1/2 :=xj x

2 . (2.3)

Here, {unj } are the computed cell averages, unj u(xj , tn)=

Ixju(, tn ) d /x ,and {(ux )nj }

are approximations to the exact derivatives, ux (xj , tn). These approximate derivatives are

reconstructed from the computed cell averages. The nonoscillatory behavior of the central

schemes hinges on the appropriate choice of approximate derivatives, and there is a library of

recipes for such nonoscillatory reconstructions. A total-variation (TV) stability, a maximum

principle, or a weaker nonoscillatory property of this piecewise-linear approximation (e.g.,decreasing the number of extrema [36, Section 4]) can be satisfied for a wide variety of

such scalar reconstructions proposed and discussed in [4, 13, 14, 27, 32, 36, 38, 41]. For

example, a scalar TVD reconstruction in (2.3) is obtained via the ubiquitous minmod limiter

[13, 31, 41],

(ux )nj = minmod

unj unj1

x,

unj+1 unjx

, (2.4)

with minmod(a, b) := 12 [sgn(a)+ sgn(b)] min(|a|, |b|). This is a particular case of a one-parameter family of limiters outlined in (5.2) below.

We then proceedto solve Eq. (2.2) subject to the piecewise-linear initial data (2.3) depicted

in Fig. 2.1.

The piecewise-linear interpolant, u(x , tn ), may be discontinuous at points {xj+1/2}. Yetfor sufficiently small t, the solution of problem (2.2)(2.3) will remain smooth around xjfor t tn+t=: tn+1, due to the finite speed of propagation. Hence, if we take Ix to be thestaggered grid cell, [xj ,xj+1] (see Fig. 2.1), we can compute u(x, t) on the RHS of (2.2)exactly, and the flux integrals there can be approximated by the midpoint rule. This resultsin the NT scheme [38]

un+1j+1/2 =unj + unj+1

2+ x

8

(ux )

nj (ux )nj+1

fun+1/2j+1 fun+1/2j , (2.5)where the midpoint values, u

n+1/2j , are predicted by Taylor expansion,

un+1/2j = unj t2 ( fx )nj . (2.6)

Thus, we have computed an approximate solution at the next time level t= tn+1, a solutionwhich is realized by its (staggered) cell averages, un+1j+1/2.

Extensions of the second order central NT scheme (2.5), (2.6) to higher-order central

schemes can be found in [4, 32, 36]. Multidimensional extensions were introduced in


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FIG. 2.1. Central differencing approachstaggered integration over the local Riemann fan.

[2, 3, 18]. We also would like to mention the corresponding central schemes for incom-

pressible flows in [2426, 33] and applications to various systems in, e.g., [1, 7, 43]. Thus,

the use of higher-order reconstructions enables us to decrease the numerical dissipation

present in central schemes, and achieve a higher resolution of shocks, rarefactions, and

other spontaneous evolution of large gradient phenomena.

Remarks. 1. Characteristic vs componentwise approach. A key advantage of central

schemes is their simplicityone avoids here the intricate and time-consuming character-

istic decompositions based on (approximate) Riemann solvers, which are necessary in

high-resolution upwind formulations. For systems of conservation laws, the numerical

derivatives (ux )nj can be implemented by componentwise extension of the scalar recipe for

nonoscillatory limiters. Similarly, the predicted values in (2.6) are based on approximate

derivatives of the flux, ( fx

)n

j. These values can be computed in terms of the exact Jacobian,

f

u(unj )(ux )

nj . Alternatively, we can even avoid the use of the computationally expensive

(and sometimes inaccessible) exact Jacobian fu

. Instead, the approximate flux derivatives,

( fx )nj , are computed in a componentwise manner based on the neighboring discrete values

of f(u(xj1, tn )), f(u(xj , tn )) and f(u(xj+1, tn )). It was pointed out in [36, 18] that thisJacobian-free version of the central scheme does not deteriorate its high resolution.

2. Cell averages vs point values. Note that here one realizes the approximate solution

by its cell averages, un+1j+1/2. In general, when dealing with first- and second-order schemes,

the cell averages, un+1j+1/2, can be identified with the corresponding point values, un+1j+1/2,modulo a negligible second-order term. We therefore from now on omit the bar notation.

(Consult [4, 36], for example, for this distinction with higher-order central schemes).

3. Second- vs first-order. In the particular case of (ux )nj 0, the second-order NT

scheme is reduced to the staggered form of the first-order LxF scheme. The nonstaggered

version of a second-order central scheme can be found in [17].


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The second-order NT scheme and its extensions owe their superior resolution to the lower

amount of numerical dissipationconsiderably lower than in the first-order LxF scheme.

The dissipation present in these central scheme has an amplitude of order O((x )2r/t);

unfortunately, this does not circumvent the difficulties with small time steps which arise, e.g.,

with convectiondiffusion equations (1.2). Consider, for example, the degenerate parabolic

equation (1.3). Stability necessitates small time steps, t (x )2, and the influence of thenumerical dissipation, accumulatedover the many steps of the NT scheme, can be clearly

seen in the smeared subshock computed in Fig. 6.22.

One possible way to overcome this difficulty is to use a semi-discrete formulation: when

a semi-discrete scheme is coupled with an appropriate ODE solver, one ends up with small

numerical viscosity proportional to the vanishing size of the time step t. But in this context,

the central LxF scheme, NT scheme, and their extensions are of limited use, since theseschemes do not admit a semi-discrete form. To make our point, consider the LxF scheme

(2.1) in its viscous form,

un+1j unjt

+ f

unj+1 f unj12x

= 12t

unj+1 unj

unj unj1. (2.7)

Passing to the limit t 0 (while leaving x to be fixed), we get the semi-discrete diver-gence on the left of (2.7), uj (t)+{f(uj+1(t)) f(uj1(t))}/2x , which is balanced withan increasing amount of dissipation on the right ux x (x)2/t, as we refine the timestep t 0. In the degenerate viscous case, for example, the CFL restriction tx isresponsible for the excessive smearing in the LxF scheme. The second-order NT scheme

has a considerably smaller numerical viscosity, with amplitude of orderO((x)4/t) away

from extrema cells.1 Nevertheless, the central NT scheme and its higher-order generaliza-

tions do not admit any semi-discrete versions, and hence are inappropriate for small time

step computations or steady-state calculations as t .This brings us to the new class of central schemes introduced in this paper. These new

central schemes have smaller numerical dissipation and are the first fully discrete Godunov-

type central schemes that admit a semi-discrete form.

3. THE FULLY DISCRETE SCHEMEONE-DIMENSIONAL SETUP

The NT scheme is based on averaging over the nonsmooth Riemann fans using spatialcells of the fixedwidth, x . The main idea in the construction of our new central schemes

is to use more precise information about the local speed of wave propagation, in order to

average the nonsmooth parts of the computed solution over smaller cells of variable size of

order O(t). We proceed as follows.

Assume that we have already computed the piecewise-linear solution at time level tn ,

based on the cell averages unj , and have reconstructed approximate derivatives (ux )nj in (2.3).

We now turn to evolve it in time. To begin with, we estimate the local speed of propagation at

the cell boundaries, xj+1/2: the upper bound (disregarding the direction of the propagation)

1 The first two terms on the right of the NT scheme (2.5) yield for smoothus

unj 2unj+1/2 + unj+12t

+ x8t

(ux )nj (ux )nj+1 (x )4

tux x x x .


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FIG. 3.2. Modified central differencing.

is denoted by anj+1/2 and is given by2

anj+1/2 = maxuC(u

j+1/2,u+j+1/2)

f

u(u)

, (3.1)

where u+j+1/2 := unj+1 x2 (ux )nj+1 and uj+1/2 := unj + x2 (ux )nj are the correspondent leftand right intermediate values ofu(x, tn ) at xj+1/2, and C(uj+1/2, u

+j+1/2) is a curve in phase

space connecting uj+1/2 and u+j+1/2 via the Riemann fan.

Remark. In most practical applications, these local maximal speeds can be easily eval-

uated. For example, in the genuinely nonlinear or linearly degenerate case one finds that

(3.1) reduces to

anj+1/2 := max

f

u

uj+1/2

,

f

u

u+j+1/2

. (3.2)

In fact, the maximal local speeds are related to the already calculated CFL number. We

emphasize that these local speeds are the only additional information required to modify

the NT scheme.

Our new scheme is constructed in two steps. First, we proceed along the lines of the

NT scheme. The NT scheme is based on averaging over the staggered control volumes[xj ,xj+1] [tn , tn+1] of fixed spatial width x . Instead, we now use narrower control vol-umes, where at each time step we integrate over the intervals [x nj+1/2,l ,x

nj+1/2,r] [tn, tn+1];

see Fig. 3.2. Due to the finite speed of propagation, the points x nj+1/2,l :=xj+1/2 anj+1/2,tand x nj+1/2,r :=xj+1/2+ anj+1/2t separate between smooth and nonsmooth regions, and

2 Let i (A) be the eigenvalues of A; then we use (A) :=maxi |i (A)| to denote its spectral radius.


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hence the nonsmooth parts of the solution are contained inside these narrower control

volumes of spatial width 2anj+1/2t.We proceed with the exact evaluation of the new cell averages at tn+1. Let xj+1/2 :=

x nj+1/2,r

x nj+1/2,l

denote the width of the Riemann fan which originates at xj+1/2

. Exact

computation of the spatial integrals yields

1

xj+1/2

x nj+1/2,r

x nj+1/2,l

u(, tn+1) d

=1

xj+1/2

x nj+1/2,r

x nj+1/2,l

u(, tn

) d 1

xj+1/2

tn+1tn

f

ux

nj+1/2,r,

fux nj+1/2,l , d= u

nj + unj+1

2+ x a

nj+1/2t

4

(ux )

nj (ux )nj+1

12anj+1/2t

tn+1tn

f

u

x nj+1/2,r,

f

u

x nj+1/2,l ,

d. (3.3)

Similarly, let xj :=x nj+1/2,l x nj1/2,r = x t(anj1/2 + anj+1/2) denote the width ofstrip around xj which is free of the neighboring Riemann fans. Then exact integration yields

1

xj

x nj+1/2,l

x nj1/2,r

u(, tn+1) d

= 1xj

x nj+1/2,lx n

j1/2,r

u(, tn ) d 1xj

tn+1tn

f

ux nj+1/2,l ,

d fux nj1/2,r, d

= unj +t

4

anj1/2 anj+1/2

(ux )

nj

1

xj

tn+1

tn

fux nj+1/2,l

, fux nj1/2,r, d. (3.4)Using the midpoint rule to approximate the flux integrals on the RHS of (3.3) and (3.4), we

conclude with the new cell averages at t= tn+1,

wn+1j+1/2 =unj + unj+1

2+ x a

nj+1/2t

4

(ux )

nj (ux )nj+1

12anj+1/2

f

un+1/2j+1/2,r fun+1/2j+1/2,l,

(3.5)

wn+1j = unj +t

2

anj1/2 anj+1/2

(ux )

nj

1 anj1/2 + anj+1/2

f

un+1/2j+1/2,l

fun+1/2j1/2,r.


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Here, the midpoint values are obtained from the corresponding Taylor expansions,

un+1/2j

+1/2,l :

=unj

+1/2,l

t

2

funj+1/2,lx , unj

+1/2,l :

=unj

+x(ux )

nj

1

2 anj

+1/2,

(3.6)

un+1/2j+1/2,r := unj+1/2,r

t

2f

unj+1/2,r

x, unj+1/2,r := unj+1x(ux )nj+1

1

2 anj+1/2

.

Again, if we want to, we can avoid the computation of the Jacobian of f while using a

componentwise evaluation of fx on the right of (3.6).

At this stage, we realize the solution at time level t

=tn+1 in terms of the approximate cell

averages, wn+1j+1/2, wn+1j . These averages spread over a nonuniform grid which is oversampled

by twice the number of the original cells at t= tn. In the second and final step of theconstruction of our scheme, we convert these nonuniform averages back into the original

grid we started with at t= tn , along the lines of the conversion recipe outlined in [17]. Asa by-product of this conversion, we avoid the staggered form of the original NT scheme

(2.5)(2.6).

To obtain the cell averages over the original grid of the uniform, nonstaggered cells

[xj1/2,xj+1/2], we consider the piecewise-linear reconstruction over the nonuniform cellsat t= tn+1, and following [17], we project its averages back onto the original uniform grid.Note that we do not need to reconstruct the average of the smooth portion of the solution,

wn+1j , as it will be averaged out (consult Fig. 3.2), and hence the required piecewise-linearapproximation takes the form

w(x, tn+1) :=j

wn+1j+1/2 + (ux )n+1j+1/2x xj+1/2

1[x n

j+1/2,l ,xnj+1/2,r]

+wn+1j 1[x nj1/2,r,x nj+1/2,l ]

. (3.7)

Here, the exact spatial derivatives, ux (xj+1/2, tn+1), are approximated by

(ux )n+1j+1/2 =

2

xminmod

wn+1j+1 wn+1j+1/2

1+ anj+1/2 anj+3/2

,wn+1j+1/2 wn+1j

1+ anj+1/2 anj1/2

. (3.8)

Finally, the desired cell averages, un+1j , are obtained by averaging the approximate solutionin (3.7). Our fully discrete second-ordercentral scheme then recasts into the final form

un+1j =1

x

xj+1/2xj1/2

w(, tn+1) d = anj1/2wn+1j1/2 +

1 anj1/2 + anj+1/2wn+1j+ anj+1/2 wn+1j+1/2 + x2

anj1/2

2(ux )n+1j1/2

anj+1/2

2(ux )n+1j+1/2

, (3.9)

where the intermediate values ofwn+1j+1/2 and wn+1j specified in (3.5) are expressed in terms

of the local speed, anj1/2, the midvalues, un+1/2j+1/2,l , u

n+1/2j+1/2,r, and the reconstructed slopes,

(ux )n+1j1/2, given in (3.2), (3.6), and (3.8), respectively.


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Remarks. 1. Central differencing. The approach taken here can be still viewed as cen-

tral differencing in the sense that the Riemann fans are inside the domain of averaging.

Consequently, since no (approximate) Riemann solvers are involved, we retain one of the

main advantages of the central schemessimplicity; no Jacobians and charactristic decom-

positions are needed. At the same time, treating smooth and nonsmooth regions separately,

we gain smaller numerical viscosity (independent ofO(1/t)). In particular, the result-

ing central scheme (3.9), (3.5) admits a semi-discrete form which is discussed in the next

section.

2. Nonoscillatory properties. The exact entropy evolution operator associated with the

scalar equation satisfies the TVD property, u(, t)BV u(, 0)BV. The various ingredi-ents in the construction of our central scheme retain this TVD propertythe nonoscillatory

reconstruction (with appropriate choice of approximate derivatives), exact evolution, andcell averaging. Thus, the only ingredient that is potentially oscillatory enters when we use

the midpoint quadrature rule for temporal integration of the fluxes, yet this does not seem

to violate the overall TVD property of our fully discrete central scheme; see, e.g., the TVD

proof of the original NT scheme in [38].

In the particular semi-discrete case (discussed in Section 4 below), the midpoint rule is

exact, and the TVD of the semi-discrete version of our scheme follows. A direct proof for

the semi-discrete scalar TVD is outlined in Theorem 4.1 below. Moreover, when this semi-

discrete scheme is coupled with appropriate RungeKutta solvers, we arrive at fully discrete,

second-order, central TVD schemes; consult Section 5 below. A maximum principle for

these schemes in two-space dimensions is outlined in Theorem 5.1.

3. Nonstaggered reconstruction. The piecewise linear reconstruction, (3.7), is neces-

sary in order to ensure second-order accuracy, since simple averaging (without reconstruc-

tion) over [xj1/2,xj+1/2] reduces the order of the resulting scheme to first-order accuracy;see [17].

4. First-order version. We conclude by commenting on the first-order version of ourscheme. To this end, we set the slopes, both (ux )

nj and (ux )

n+1j+1/2, to be zero. Then the

staggered cell averages in (3.5) are reduced to

wn+1j+1/2 =unj + unj+1

2 1

2anj+1/2

f

unj+1 funj, wn+1j = unj .

Inserting these values into (3.9) yields the first-order scheme, which takes the viscous

form

un+1j = unj

2

f

unj+1 funj1+ 12

anj+1/2

unj+1 unj

anj1/2unj unj1 ,(3.10)

where anj+1/2 are the maximal local speeds. This scheme was originally attributed to Rusanov[30]; it is a special case of the family of first-order Godunov-type scheme introduced in [15]

(based on a symmetric approximate Rieman solver) and it coincides with the so-called localLxF scheme in [45]. It should be noted, however, that although this scheme is similar to the

LxF scheme, (2.7), its numerical viscosity coefficient, Qnj+1/2 = anj+1/2, is always smallerthan the corresponding LxF one, QLxFj+1/2 1, thanks to the CFL condition. In regions witha small local speed of propagation (e.g., near the sonic points), the numerical viscosity

present in (3.10) is in fact considerably smaller, tanj+1/2x .


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4. THE REDUCTION TO SEMI-DISCRETE FORMULATION

4.1. One-Dimensional Hyperbolic Conservation Laws

We consider the fully discrete second-order central scheme (3.9), (3.5), expressed interms of the cell averages, wn+1j1/2, w

n+1j , and the approximate derivates, (ux )

n+1j1/2. Observe

that except the original averages, unj , which participates in wnj , all the terms on the right of

(3.9), (3.5) are proportional to t (or ). Rearranging the divided differences accordingly

while separating the vanishing terms proportional to (as t 0) we find

un+1j unjt

(3.9)

=anj1/2

xwn+1j1/2 +

1

x a

nj1/2 + anj+1/2

x

wn+1j +

anj+1/2x

wn+1j+1/2 1

tunj +O()

(3.5)

=

anj1/22x

unj1+ unj

+ 14

anj1/2

(ux )nj1(ux )nj

12x

f

un+1/2j1/2,r

fun+1/2j1/2,l

anj1/2 + anj+1/2

xunj +

1

2anj1/2 anj+1/2

(ux )

nj

1

x f

u

n+1/2j+1/2,l

fun+1/2j1/2,r+ anj+1/22x

unj + unj+1

+ 14

anj+1/2

(ux )nj (ux )nj+1

1

2x

f

un+1/2j+1/2,r

fun+1/2j+1/2,l+O()

= 12x

fun+1/2j+1/2,r+ fun+1/2j+1/2,l fun+1/2j1/2,r+ fun+1/2j1/2,l

+an

j+1/2x

unj+1 x2 (ux )nj+1

unj + x2 (ux )nj

anj1/2x

unj

x

2(ux )

xj

unj1 +x

2(ux )

nj1

.

+O().

Note that as t 0, the midvalues on the right approach (consult (3.6))

un+1/2j+1/2,r uj+1(t)x

2(ux )j+1(t) =: u+j+1/2(t),

(4.1)

un+1/2j+1/2,l uj (t)+

x

2(ux )j (t) =: uj+1/2(t),

where (ux )j (t) are the numerical derivatives reconstructed from the computed cell averages,

uj (t). Thus, letting t 0, the resulting semi-discrete central scheme can be written in itscompact form

d

dtuj (t) =

f

u+j+1/2(t)+ fuj+1/2(t) fu+j1/2(t)+ fuj1/2(t)

2x

+ 12x

aj+1/2(t)

u+j+1/2(t) uj+1/2(t)

aj1/2(t)u+j1/2(t) uj1/2(t) .(4.2)


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Recall that aj+1/2(t) is the maximal local speed, e.g., in the generic case one may take

aj+1/2(t) :=maxf

u u+j+1/2(t),

f

u uj+1/2(t).

Remarks. 1. Conservation form.The second-order scheme, (4.2), admits the conserva-

tive form,

d

dtuj (t) = Hj+1/2(t) Hj1/2(t)

x, (4.3)

with the numerical flux

Hj+1/2(t) :=fu+j+1/2(t)+ fu

j

+1/2(t)

2 aj+

1/2(t)

2

u+j+1/2(t) uj+1/2(t). (4.4)Here, the intermediate values uj+1/2 are given by

u+j+1/2 := uj+1(t)x

2(ux )j+1(t), uj+1/2 := uj (t)+

x

2(ux )j (t). (4.5)

One verifies that Hj+1/2(t) H(uj1(t), uj (t), uj+1(t), uj+2(t)) is a numerical flux con-sistent with Eq. (1.1), i.e., H(v,v,v,v) = f(v). In fact, with the minmod limiter, (2.4),the corresponding approximate derivatives, (ux )j (t), vanish at extrema values,

sgn(uj+1(t) uj (t))+ sgn(uj (t) uj1(t)) = 0 (ux )j (t) = 0, (4.6)and hence the corresponding numerical flux satisfies the essentially three-pointconsistency

H(, v , v , ) = f(v). (4.7)2. Numerical viscosity. The second expression on the right of (4.2) accounts for the

numerical viscosity of the scheme. Taylors expansion shows that for smooth us, this amountof numerical viscoity is of order (x )3(a(u)uxxx )x /8. This O(x )3 term, uniformlybounded w.r.t. 1/t, should be contrasted with the corresponding numerical viscosity terms

of order O((x )2/t) in the first-order LxF scheme (indicated earlier in (2.7)) and of order

O((x)4/t) in the second-order NT scheme.

3. Simplicity. We again would like to emphasize the simplicity of the second-order

semi-discrete central scheme, (4.2), so that it does not require any information about the

eigenstructure of the underlying problem beyond the CFL-related speeds, aj+

1/2(t). The

computation of the numerical derivatives, (ux )j (t),iscarriedout componentwise; no specificknowledge of characteristic decomposition based on (approximate) Riemann solvers is

required.

4. First-order reduction. If we reset all the numerical derivatives, (ux )j (t)= 0, then(4.2) is reduced to the first-order semi-discrete central scheme corresponding to the t 0-limit of the Rusanov scheme, (3.10),

d

dtuj (t) = f(uj

+1(t))

f(uj

1)

2x

+ 12x

aj+1/2(t)(uj+1(t) uj (t)) aj1/2(t)(uj (t) uj1(t))

. (4.8)

We conclude this section with the proof of the one-dimensional scalar TVD property for

our new semi-discrete scheme.


14/42


THEOREM 4.1 (TVD of Semi-discrete Central Scheme). Consider the scalar semi-

discrete central scheme (4.2), with intermediate values uj1/2 in (4.5) based on approximatederivatives (ux )j (t) satisfying (4.6), e.g., the family of minmod-like limiters (with = 1 cor-responding to (2.4))

(ux )nj := minmod

unj unj1x

,unj+1 unj1

2x,

unj+1 unjx

, 1 2. (4.9)

Then the following TVD property holds:

u(, t)BV :=j|uj+1(t) uj (t)| u(, 0)BV.

Remark. Notice that in the scalar case the local propagation speeds are given by

aj+1/2(t) := maxu[u

j+1/2(t),u+j+1/2(t)]

|f(u)|, (4.10)

and in the special case of convex f, this is further simplified:

aj+1/2(t) := max{|f(uj+1/2(t))|, |f(u+j+1/2(t))|}. (4.11)

Proof. The second-order flux in (4.2), Hj+1/2(t),canbeviewedasageneralizedMUSCLflux [40],

Hj+1/2(t)= HRus

u+j+1/2(t), uj+1/2(t)

,

expressed in terms of the first-order E-flux HRus

= HRus

(u, ur), associated with the first-order Rusanov scheme (4.8),

HRus(u, ur) := f(u)+ f(ur)2

ar2

(ur u), ar= maxu[u,ur]

|f(u)|.

According to [47], the TVD property of such scalar, semi-discrete generalized MUSCL

schemes is guaranteed if (consult [47, Example 2.4])

(ux )j+1/2uj1/2 2. (4.12)

This is clearly fulfilled by the choice of approximate derivatives in (4.9). (We note in passing

the necessity of the clipping phenomenon, (4.6), enforced by (4.12).)

4.2. One-Dimensional ConvectionDiffusion EquationsConsider the convectiondiffusion equation (1.2). If the dissipation flux, Q(u, ux ), is a

nonlinear function, then Eq. (1.2) can be a strongly degenerate parabolic equation which

admits nonsmooth solutions. To solve it numerically is a highly challenging problem. In

this context, the operator splitting technique was used in, e.g., [6, 8, 12, 20, 22, 23], yet this

approach suffers the familiar limitations of splitting, e.g., limited accuracy, etc.


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Our second-order semi-discrete scheme, (4.3)(4.4), can be applied to Eq. (1.2) in a

straightforward manner, since we can treat the hyperbolic and the parabolic parts of (1.2)

simultaneously. This results in the following conservative scheme:

uj (t)=Hj+1/2(t) Hj1/2(t)x

+ Pj+1/2(t) Pj1/2(t)x

. (4.13)

Here, Hj+1/2(t) is our numerical convection flux, (4.4), and Pj+1/2(t) is a reasonable ap-proximation to the diffusion flux, e.g., the simplest central difference approximation

Pj+1/2(t)= 12

Q

uj (t),

uj+1(t) uj (t)x

+ Q

uj+1(t),

uj+1(t) uj (t)x

. (4.14)

4.3. Multidimensional Extensions

Our second-order semi-discrete schemes, (4.2) and (4.13), (4.4), (4.14), can be extended

to both multidimensional hyperbolic and parabolic problems. Without loss of generality, let

us consider the two-dimensional convectiondiffusion equation

ut+ f(u)x + g(u)y = Qx (u, ux , uy )x + Qy (u, ux , uy )y , (4.15)where Qx Qy 0 corresponds to the two-dimensional hyperbolic conservation law.

We use a uniform spatial grid, (xj , yk)= (j x, ky). Suppose that we have computedthe solution at some time level tand have reconstructed the two-dimensional, non-oscillatory

piecewise-linear polynomial approximation

u(x , y, t) j,k

[uj,k(t)+(ux )j,k(t)(xxj )+(uy )j,k(t)(yyk)]1[xj1/2,xj+1/2] [yk1/2,yk+1/2].

Here, xj1/2 :=xj x2 , yk1/2 :=yk y2 ; (ux )j,k(t) and (uy )j,k(t) are numerical deriva-tives, which approximate the exact ones, ux (xj , yk, t) and uy (xj , yk, t), respectively. With

a proper choice of numerical derivatives, the reconstruction of piecewise polynomial ap-

proximation is nonoscillatory. For example, using the minmod limiter, (2.4), guarantees

the nonoscillatory property in the sense of satisfying a (local) scalar maximum principle;

consult [18, Thm. 1].

The 2D extension of the scheme (4.13), (4.4), (4.14) can be written in the conservative

form

d

dtuj,k(t) =

Hxj+

1/2,k

Hxj

1/2,k

x H

yj,k+

1/2

H

yj,k

1/2

y

+ Pxj+1/2,k Pxj1/2,k

x+ P

yj,k+1/2 Pyj,k1/2

y. (4.16)

Here, Hxj+1/2,k Hxj+1/2,k(t) and Hyj,k+1/2 Hyj,k+1/2(t) are x- and y-numerical convectionfluxes, respectively (viewed as a generalization of the one-dimensional flux constructed

above in (4.4)),

Hxj+1/2,k(t) :=f

u+j+1/2,k(t)+ fuj+1/2,k(t)

2 a

xj+1/2,k(t)

2

u+j+1/2,k(t) uj+1/2,k(t)

,

(4.17)

Hyj,k+1/2(t) :=

g

u+j,k+1/2(t)+ guj,k+1/2(t)

2 a

yj,k+1/2(t)

2

u+j,k+1/2(t) uj,k+1/2(t)

,


16/42


which are expressed in terms of the intermediate values

uj+1/2,k(t) := uj+1,k(t)x

2(ux )j+1/21/2,k(t),

(4.18)uj,k+1/2(t) := uj,k+1(t)

y

2(uy )j,k+1/21/2(t),

and the local speeds, axj+1/2,k(t) and ayj,k+1/2(t), are computed, e.g., by

axj+1/2,k(t) := max

f

u

uj+1/2,k(t)

, a

yj,k+1/2(t) := max

g

u

uj,k+1/2(t)

.

(4.19)

Similarly, Pxj+1/2,k Pxj+1/2,k(t) and Pyj,k+1/2 Pyj,k+1/2(t) are the corresponding x- andy-numerical diffusion fluxes, given by

Pxj+1/2,k :=1

2

Qx

uj,k,

uj+1,k uj,kx

, (uy )j,k

+ Qxuj+1,k,uj+1,k uj,k

x, (uy )j+1,k,

(4.20)P

yj,k+1/2 :=

1

2

Qy

uj,k, (ux )j,k,

uj,k+1 uj,ky

+ Qy

uj,k+1, (ux )j,k+1,uj,k+1 uj,k

y

.

5. FROM SEMI-DISCRETE BACK TO FULLY DISCRETETHE GENERAL SETUP

The two-dimensional semi-discrete central scheme (4.2) forms a system of nonlinear

ODEs, the so-called method of lines for the discrete unknowns {uj,k(t)}. To integratein time, one must introduce a variable time step, tn , stepping forward from time level

tn to tn+1 := tn + tn. We start by considering the simplest scenario of first-order timedifferencing. The nonoscillatory behavior of the forward Euler scheme is summarized in

the maximum principle stated in Theorem 5.1 below. To retain the overall high accuracy of

the spatial differencing, however, higher-order stable time discretizations are required. To

this end, the forward Euler time differencing can be used as a building block for higher-order RungeKutta and multi-level ODE solvers. In particular, second- and third-order ODE

solvers can be constructed by convex combinations of the simple forward Euler differencing,

retaining the overall maximum principle. Thus, we conclude in Corollaries 5.1 and 5.2

below with a fully discrete second-order central scheme satisfying the two-dimensional

scalar maximum principle.

We begin with

THEOREM 5.1 (Maximum Principle). Consider the two-dimensional central scheme

un+1j,k = unj,ktn

x

Hxj+1/2,k(t

n ) Hxj1/2,k(tn ) tn

y

H

yj,k+1/2(t

n)Hyj,k1/2(tn )

.

(5.1)

Here, Hx (t) and Hy (t) are the numerical fluxes given in (4.17)(4.19); let their numerical

derivatives be determined by one of the following one-parameter family of minmod-like


17/42


limiters:

(ux )nj,k := minmod

unj,k unj1,k

x,

unj+1,k unj1,k2x

, unj+1,k unj,k

x , 1 2,

(5.2)

(uy )nj,k := minmod

unj,k unj,k1y

,uj,k+1 uj,k1

2y,

unj,k+1 unj,ky

, 1 2.

Assume the following CFL condition holds:

max

tn

xmax

u|f(u)|, t

n

ymax

u|g(u)|

1

8. (5.3)

Then the resulting fully discrete central scheme satisfies the maximum principle

maxj,k{uj,k(tn+1)} max

j,k{uj,k(tn )}. (5.4)

Proof. With n := tn/x and n := tn /y denoting the x- and y-mesh ratios, theforward Euler time discretization of our scheme takes the explicit form

un+1j,k=

unj,k

n

2 fu+j+1/2,k(tn )+

fuj+1/2,k(tn )

fu+j1/2,k(tn )

fuj1/2,k(tn )

axj+1/2,k(tn )

u+j+1/2,k(tn ) uj+1/2,k(tn )

+ axj1/2,k(tn )u+j1/2,k(tn ) uj1/2,k(tn )

n2

g

u+j,k+1/2(tn )+ guj,k+1/2(tn) gu+j,k1/2(tn )

guj,k1/2(tn ) ayj,k+1/2(tn)u+j,k+1/2(tn ) uj,k+1/2(tn)+ ayj,k1/2(tn )

u+j,k1/2(t

n) uj,k1/2(tn )

.

To simplify notations, we use the standard abbreviations

xj1/2,ku := u+j1/2,k(tn)uj1/2,k(tn ), xj,k f := f

uj+1/2,k(tn ) fu+j1/2,k(tn),

with the similar notations for y-differences, e.g., yj,kg := g(uj,k+1/2(tn ))g(u+j,k1/2(tn )),

etc. Then our scheme can be rewritten as follows (where all the quantities on the right are

taken at time level t= tn ):

un

+1

j,k =uj+

1/2,k

+u+j

1/2,k

+uj,k

+1/2

+u+j,k

1/2

4 n

2xj+1/2,k fxj+1/2,ku u+j+1/2,k uj+1/2,k

+ 2 xj,k f

xj,ku

uj+1/2,k u+j1/2,k

+ xj1/2,k fxj1/2,ku

u+j1/2,k uj1/2,k

axj+1/2,k

u+j+1/2,k uj+1/2,k+ axj1/2,ku+j1/2,k uj1/2,k

n

2yj,k+1/2g

yj,k+1/2u

u+j,k+1/2 uj,k+1/2+ 2

yj,kg

yj,ku

uj,k+1/2 u+j,k1/2+

yj,k1/2g

yj,k1/2u

u+j,k1/2 uj,k1/2

ayj,k+1/2u+j,k+1/2 uj,k+1/2

+ ayj,k1/2

u+j,k1/2 uj,k1/2

.


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Rearranging the terms, we find that un+1j,k is given by the following linear combination ofthe intermediate values, uj1/2,k and u

j,k1/2:

un+1j,k = n

2

axj+1/2,k

x

j+1/2,k fxj+1/2,ku

u+j+1/2,k+

n

2

axj1/2,k+

x

j1/2,k fxj1/2,ku

uj1/2,k

+

1

4+

n

2

xj+1/2,k fxj+1/2,ku

axj+1/2,k 2xj,k f

xj,ku

uj+1/2,k

+

1

4

n

2

xj+1/2,k fxj+1/2,ku

+ axj1/2,k 2xj,k f

xj,ku

u+j1/2,k

+ n

2

ayj,k+1/2

y

j,k+1/2g

yj,k+1/2u

u+j,k+1/2 +

n

2

ayj,k1/2 +

y

j,k1/2g

yj,k1/2u

uj,k1/2

+

1

4+

n

2

yj,k+1/2g

yj,k+1/2u

ayj,k+1/2 2

yj,kg

yj,ku

uj,k+1/2

+

1

4

n

2

yj,k1/2g

yj,k1/2u

+ ayj,k1/2 2

yj,kg

yj,ku

u+j,k1/2. (5.5)

Note that all the coefficients in (5.5) are positive due to our CFL assumption, (5.3). Thismeans that the linear combination on the RHS of (5.5) is a convex combination and hence

the value of un+1j,k does not exceed the values of uj1/2,k and u

j,k1/2. And since our

choice of minmod-like approximate derivatives in (5.2) guarantees that these intermedi-

ate values, u, satisfy a local maximum principle w.r.t. the original averages, un , e.g., [18],maxj,k{uj1/2,k(t), uj,k1/2(t)} maxj,k{uj,k(t)}, the result (5.4) then follows.

The forward Euler scheme is limited to first-order accuracy. It can be used, however, as

a building block for higher-order schemes based on RungeKutta (RK) or multi-level timedifferencing. Shu and Osher [44, 45] have identified a whole family of such schemes, based

on convex combinations of forward Euler steps.

To this end, we let C[w] denote our spatial recipe (4.17)(4.19) for central differencing

a grid function w= {wj,k},

C[w] :

= Hxj+1/2,k(w) Hxj1/2,k(w)

x +

Hyj,k+1/2(w) Hyj,k1/2(w)

y. (5.6)

Expressed in terms of the forward Euler solver, w+tC[w], we consider the one-parameterfamily of RK schemes

u(1) = un +tnC[un ]u(+1) = un + (1 )

u() +tn C

u()

, = 1, 2, . . . , s 1, (5.7)

un

+1 :=

u(s).

In Table 5.1 we quote the preferred second- and third-order choices of [45]. We state

COROLLARY 5.1 (Maximum Principle for RungeKutta Time Differencing). Assume

that the CFL condition (5.3) holds. Then the fully discrete central scheme (4.17)(4.19),

(5.2),(5.7) with m specified in Table 5.1 satisfies the maximum principle (5.4).


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TABLE 5.1

RungeKutta Methods

1 2

Second order time differencing

Two-step modified Euler (s= 2) 12

Third order time differencing

Three-step method (s= 3) 34

1

3

A similar two-parameter family ofmulti-level methods was identified in [44]. They take

the particularly simple form

un+1 = (un + c0tnC[un ])+ (1 )(uns + cs tnC[uns ]), (5.8)with positive coefficients given in Table 5.2. We state

COROLLARY 5.2 (Maximum Principle for Multi-level Time Differencing). Assume that

the CFL condition

maxtn

x

maxu |

f(u)|,

tn

y

maxu |

g(u)| minck

1

8ck(5.9)

holds. Then the fully discrete multi-level central scheme (4.17)(4.19), (5.2), (5.8), with

and cs specified in Table 5.2, satisfies the maximum principle (5.4).

We close by noting that Corollaries 5.1 and 5.2 extend the maximum principle for the

second-order fully discrete two-dimensional scheme introduced in [18, Thm. 1].

6. NUMERICAL EXAMPLES

We conclude this paper with a number of numerical examples. In all the numerical

results presented below we have used the -dependent family of limiters corresponding

to (4.9). (These are in general the less dissipative limiter than the original minmod, (2.4),

corresponding to (4.9) with = 1). The spatial derivative ofu(x , t) is approximated by

ux (x , t)minmodu(x + a) u(x )

a,

u(x + a) u(x b)a

+b

, u(x) u(x b)

b ,(6.1)

TABLE 5.2

Multi-level Methods

c0 cs

Second-order time differencing

4-level method (s

=2) 3

42 0

5-level method (s= 3) 89

3

20

Third-order time differencing


3 1211


2 107


5

3

30

17


20/42


where a and b are appropriate grid scales, and the multivariable minmod function is defined

by

minmod(x1,x2, . . . )=

minj

{xj

}, if xj > 0

j,

maxj {xj }, if xj < 0 j,0, otherwise.

The parameter [1, 2] has been chosen in the optimal way in every example. Note that = 2 corresponds to the least dissipative limiter (no new local extrema are introduced),whereas = 1 ensures a nonoscillatory nature of the approximate solution in the sensethat there is no increase of the total-variation. The reconstruction depicted in Fig. 3.2, for

example, does not increase in the variation at the interface xj1/2 but it is an oscillatory onesince new extrema is introduced at xj+1/2. In the scalar examples below, = 2 has provideda satisfactory results, but consult the nonconvex BuckleyLevertt equation in Fig. 6.7 as a

counterexample; for systems the optimal values of vary between 1.1 and 1.5.

Another point we would like to stress concerns the time discretization of our semi-discrete

central schemes. In general, the RungeKutta time differencing is preferable over the multi-

level differencing, since the former enables a straightforward use of variable time steps.

We note that for the standard explicit RK methods the time step can be very small due to

their strict stability restriction. There are two different approaches to increasing efficiencyat this point. First, one can use implicit or explicitimplicit ODE solvers. These methods are

unconditionally stable, but they require inverting nonlinear operators (in the general case of

a nonlinear diffusion), which is a computationally expensive and analytically complicated

procedure.

In all the numerical examples shown below, we preferred the second approachto solve

systems of ODEs by means of the explicit embedded integration third-order RK method

recently introduced by Medovikov [37] (his original code, DUMKA3, was used). This high-

order differencing produces accurate results, and its larger stability domains (in comparison

with the standard RK methods) allow us to use larger time steps; the explicit form retains

simplicity, and the embedded formulas permit an efficient stepsize control. In practice these

methods preserve all the advantages of explicit methods and work as fast as implicit methods

(see [37] for details).

Remark. Below, we abbreviate by FD1 and SD1 the Rusanov first-order fully and

semi-discrete schemes. We also use FD2 and SD2 notation for our second-order fully and

semi-discrete schemes. As previously, LxF and NT stand for the LaxFriedrichs and the

NessyahuTadmor schemes.

6.1. One-Dimensional Scalar Linear Hyperbolic Equation

EXAMPLE 1 (Linear Steady Shocks). First, consider the simplest linear equation with

the zero flux, f(u)= 0,

ut= 0, (6.2)

subject to the discontinuous initial data,

u(x, 0)=

1, 0.5


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FIG. 6.3. Problem (6.2)(6.3). N= 100, T= 2; first-order methods.

Notice that due to their dissipative nature, neither the LxF scheme nor the NT scheme

provides a reasonable approximation to this problem. But using the Rusanov scheme and

our second-order fully discrete scheme, which are, in some sense, the least dissipative

central schemes, we achieve the perfect resolution of the discontinuities (the same result is

obtained by the corresponding semi-discrete schemes).

In Figs. 6.3 and 6.4 the solutions computed by our schemes are compared with the

(staggered) LxF and NT schemes. It also can be easily checked analytically that both the

FD1 scheme and the FD2 schemes solve problem (6.2)(6.3) exactly.

FIG. 6.4. Problem (6.2)(6.3). N= 100, T= 2; second-order methods.


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TABLE 6.1

Initial Value Problem (6.4), L1- and L-Norms of the Errors

L 1-error L-error

N NT Rate FD2 Rate NT Rate FD2 Rate

40 2.920e-03 8.716e-03 3.151e-03 7.818e-03

80 4.583e-04 2.67 1.876e-03 2.22 9.963e-04 1.66 2.598e-03 1.59

160 1.115e-04 2.04 3.892e-04 2.27 3.704e-04 1.43 9.262e-04 1.49

320 2.360e-05 2.24 7.943e-05 2.29 1.263e-04 1.55 2.881e-04 1.68

640 5.273e-06 2.16 1.659e-05 2.26 4.463e-05 1.50 1.028e-04 1.49

1280 1.249e-06 2.08 3.430e-06 2.27 1.690e-05 1.40 3.593e-05 1.52

EXAMPLE 2 (Accuracy Test). Let us consider the linear equation subject to periodic

initial data

ut+ ux = 0, u(x, 0)= sinx . (6.4)This problem admits the global smooth solution that was computed at time T= 1 with thevarying number of grid points, N.

In Table 6.1 we compare the accuracy of our second-order fully discrete scheme, FD2,with the accuracy of the NT scheme. These results show that for the FD2 scheme the absolute

error is larger, but the rate of convergence is slightly higher than for the NT scheme.

6.2. One-Dimensional Scalar Hyperbolic Conservation Laws

EXAMPLE 3 (Burgers Equation: Pre- and Post-shock Solutions). In this example we

approximate solutions to the inviscid Burgers equation,

ut+

u2

2

x

= 0. (6.5)

Let us start with the case of smooth periodic initial data, e.g.,

u(x, 0) = 0.5+ sinx . (6.6)The well-known solution of (6.5)(6.6) develops a shock discontinuity at the critical time

Tc= 1. Table 6.2 shows the L1- and L-norms of the errors at the pre-shock time T= 0.5

TABLE 6.2

Initial Value Problem (6.5)(6.6), L1- and L-Norms of the Errors at T= 0.5

L 1-error L-error

N NT Rate FD2 Rate NT Rate FD2 Rate

40 1.011e-02 9.101e-03 6.782e-03 6.283e-03

80 2.116e-03 2.26 1.843e-03 2.30 2.951e-03 1.20 2.333e-03 1.43

160 4.705e-04 2.17 4.272e-04 2.11 9.918e-04 1.57 7.481e-04 1.64

320 1.095e-04 2.10 9.334e-05 2.19 3.727e-04 1.41 2.603e-04 1.52

640 2.517e-05 2.12 2.163e-05 2.11 1.248e-04 1.58 9.508e-05 1.45

1280 5.926e-06 2.09 4.867e-06 2.15 4.433e-05 1.49 3.132e-05 1.60


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FIG. 6.5. Problem (6.5)(6.6). N= 100, T= 2; the FD2 scheme.

when the solution is still infinitely smooth. Unlike the linear case (Example 2), both the ab-

solute errors and the convergence rates of the FD2 scheme are better than the corresponding

errors and rates of the NT scheme. This indicates a certain advantage of our fully discrete

second-order scheme over the NT scheme while applied to nonlinear problems.

In Figs. 6.5 and 6.6 we present the approximate solutions at the post-shock time T= 2,when the shock is well developed. Second-order behavior is confirmed by the measuring

Lip-errors, [39], which are recorded in Table 6.3. Again, the solution obtained by the FD2

scheme is slightly more accurate than the solution computed by the NT scheme.

FIG. 6.6. Problem (6.5)(6.6). N= 100, T= 2; the NT scheme.


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TABLE 6.3

Initial Value Problem (6.5)(6.6), Lip-Norm of the Errors at T= 2

N NT Rate FD2 Rate

40 1.233e-02 7.779e-03

80 1.904e-03 2.70 1.982e-03 1.97

160 4.809e-04 1.99 4.905e-04 2.01

320 1.253e-04 1.94 1.222e-04 2.01

640 3.415e-05 1.88 3.047e-05 2.00

1280 1.000e-05 1.77 7.558e-06 2.01

EXAMPLE 4 (Nonconvex Flux). In this example we show results of applying our fully

discrete second-order scheme to the following Riemann problem:

ut+

(u2 1)(u2 4)4

x

= 0, u(x, 0) =

2, if x < 0,

2, ifx > 0. (6.7)

The solutions to this initial value problem are depicted at time T= 1.2. Figure 6.7 demon-strates the clear advantage of our new FD2 scheme over the NT scheme; in particular, thelatter seems to give a wrong solution (even after the grid refinement.3 We note, however,

that when a more restrictive minmod limiter was used (corresponding to = 1 in (6.1)), theNT solution did converge to the entropy solution at the expense of additional smoothing on

the edges of the Riemann fan, which can be noticed in Fig. 6.8.

6.3. One-Dimensional Systems of Hyperbolic Conservation Laws

EXAMPLE 5 (Euler Equations of Gas Dynamics). Let us consider the one-dimensionalEuler System

t

m

E

+

x

m

u2 + pu(E+ p)

= 0, p = ( 1) E

2u2

,

where , u, m

=u, p, and E are the density, velocity, momentum, pressure, and total

energy, respectively. Here, the conserved quantities are u= (, m, E)T, and the flux vectorfunction is f(u)= (m, u2+ p, u(E+ p))T. We solve this system subject to Riemanninitial data,

u(x , 0) = uL , x < 0,uR , x > 0.

We apply our scalar-designed schemes to this problem in a straightforward manner. We now

prefer the alternative approximation to the flux derivatives, needed in (3.6). The minmodlimiter, (6.1), is employed directly on the corresponding values of f(u) to avoid an expensivecomputation of the Jacobian, f

u.

3A similar failure of convergence towards the entropy solution by upwind approximation of nonconvex equations

was reported in [5].


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FIG. 6.7. Riemann IVP (6.7). N= 100.

We compute the solution to two different Riemann problems:

The first Riemann problem was proposed by Sod [46]. The initial data are givenby

uL = (1, 0, 2.5)T, uR = (0.125, 0, 0.25)T.

The approximations to the density, velocity, and pressure obtained by the FD2 scheme arepresented in Figs. 6.96.14.

FIG. 6.8. Riemann IVP (6.7). N= 100.


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FIG. 6.9. Sod problemdensity. N= 200, T= 0.1644.

FIG. 6.10. Sod problemdensity. N= 400, T= 0.1644.

FIG. 6.11. Sod problemvelocity. N= 200, T= 0.1644.


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FIG. 6.12. Sod problemvelocity. N= 400, T= 0.1644.

FIG. 6.13. Sod problempressure. N= 200, T= 0.1644.

FIG. 6.14. Sod problempressure. N= 400, T= 0.1644.


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FIG. 6.15. Lax problemdensity. N= 200, T= 0.16.

The second Riemann problem was proposed by Lax [29]. The initial values aregiven by

uL = (0.445, 0.311, 8.928)T, uR = (0.5, 0, 1.4275)T.

The results computed by the FD2 scheme are shown in Figs. 6.156.20.

Our numerical results for this system are comparable with the results obtained in

[38]. We would like to stress again that as in the case of the original NT scheme the

characteristic decomposition is not required; i.e., our new schemes still can be applied

componentwise.

Remark. In all the above 1D hyperbolic examples we have presented the numerical

results obtained by our fully discrete scheme. We also tested the corresponding

FIG. 6.16. Lax problemdensity. N= 400, T= 0.16.


29/42


FIG. 6.17. Lax problemvelocity. N= 200, T= 0.16.

FIG. 6.18. Lax problemvelocity. N= 400, T= 0.16.

FIG. 6.19. Lax problempressure. N= 200, T= 0.16.


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FIG. 6.20. Lax problempressure. N= 400, T= 0.16.

semi-discrete scheme on the same examples. The results are very similar, but the semi-

discrete scheme is slightly more dissipative than the fully discrete one.

As we have already mentioned, the main advantage of the semi-discrete approach can beseen while we apply our scheme to (degenerate) parabolic convectiondiffusion equations.

Below, we show several examples of such problems.

6.4. One-Dimensional ConvectionDiffusion Equations

EXAMPLE 6 (Burgers-Type Equation with Saturating Dissipation). We begin with the

convectiondiffusion equation with bounded dissipation flux proposed in [28]. Consider

Eq. (1.3) with f(u)= u2 subject to the Riemann initial data,

u(x, 0) =

1.2, x < 0,

1.2, x > 0. (6.8)

It was proved in [12] that the solution to this initial value problem contains a subshock

located at x = 0. This is why solving (1.3), (6.8) numerically is quite challenging problem.Our second-order semi-discrete scheme, SD2, provides a very good resolution of the

discontinuity (Fig. 6.21) while the fully discrete NT scheme fails to resolve it (Fig. 6.22;see also numerical results in [28]). The SD2 scheme was tested on all the examples from

[12]. The numerical results are highly satisfactory, and using this semi-discrete approach

no operator splitting is needed (consult [12] for details).

EXAMPLE 7 (BuckleyLeverett Equation). Next, let us consider the scalar convection

diffusion BuckleyLeverett equation (1.2) with Q(u, s)= (u)s,

ut+ f(u)x = ((u)ux )x , (u) 0. (6.9)This is a prototype model for oil reservoir simulations (two-phase flow). Typically, (u)

vanishes at some values of u, and (6.9) is a degenerate parabolic equation. Usually, the

operator splitting technique is used (see [6, 8, 20, 22, 23]) to solve it numerically, but the

limitations of such an approach are well known.


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FIG. 6.21. Sharp resolution by the SD2 scheme. N= 400, T= 1.5.

We take to be 0.01, f(u) to have an s-shaped form,

f(u) = u2

u2 + (1 u)2 , (6.10)and (u) to vanish at u= 0, 1,

(u)

=4u(1

u). (6.11)

FIG. 6.22. Smeared discontinuity by the NT scheme. N= 400, T= 1.5.


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FIG. 6.23. Initial-boundary value problem (6.9)(6.12).T= 0.2.

The initial function is

u(x, 0) =

1 3x , 0x 13

,

0, 13


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FIG. 6.24. Riemann problem for the BuckleyLeverett equation with and without gravitation.T= 0.2.

EXAMPLE 9 (Glacier Growth Model). In this example we consider a one-dimensional

model for glacier growth (see [9, 21]). Let a glacier of height h(x , t) rest upon a flat

mountain. Its evolution is described by the nonhomogenious convectiondiffusion equation

ht+ f(h)x = ((h)hx )x + S(x, t, h). (6.14)

Let = 0.01. The typical flux and diffusion coefficient are

f(h) = h + 3h6

4, (h) = 3h6. (6.15)

We first look at the Riemann problem with

h(x, 0) = 1, x < 0,0, x > 0,that describes an outlet into a valley disregarding seasonal variations. To complete this simple

model we use the source S(x , t, h)= S0(x) if h(x, t) > 0, and S(x , t, h)=max{S0(x), 0}ifh(x, t)= 0, where

S0(x) = 0, x 0.2.

The numerical simulations for different number of grid points and at different times are

presented in Figs. 6.256.27. These solutions, obtained by our SD2 scheme, seem to be

more accurate than the solutions obtained by the operator splitting method in [21].


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FIG. 6.25. Moving glacier at T= 1.




35/42


FIG. 6.28. Growing glacier at T= 7.5.

Second, we look at the growth of a new glacier (h(x, 0) 0). Let the glacier be restrictedto the interval [5, 5]. Then an appropriate source term is

S(x, t) =

0, x 5,0.01x + 0.05 sin(2 t), x >5.

Here the second, trigonometric term models seasonal variations.

Figures 6.286.33 illustrate the glacier growth computed by the Rusanov first-order andby our second-order semi-discrete schemes, SD1 and SD2, with different numbers of grid



36/42


FIG. 6.30. Growing glacier at T= 15.

points. The SD2 scheme provides more accurate resolution of the upstream front, but admits

some oscillations on the glacier downstream. The amplitude of these oscillations remains

small but does not diminish with the grid refinement. Moreover, they tend to propagate up

to the top of the glacier as N increases. At the same time, applying the SD1 scheme with a

large number of grid points gives a very accurate, nonoscillatory solution, comparable with

the one reported in [21].

EXAMPLE 10 (HyperbolicParabolic Equation). We conclude this subsection with anexample of strongly degenerate parabolic (or, hyperbolicparabolic) convectiondiffusion



37/42


FIG. 6.32. Growing glacier at T= 30.

equation. Consider Eq. (6.9) with = 0.1, f(u)= u2, and

(u) =

0, |u| 0.25,1, |u|> 0.25. (6.16)

This (u) is a discontinuous function, and the equation is therefore of hyperbolic nature

when u [0.25, 0.25] and parabolic elsewhere.



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FIG. 6.34. Initial value problem (6.9), (6.16), (6.17). T= 0.7.

We apply our SD2 scheme to Eqs. (6.9), (6.16) subject to the initial data,

u(x, 0) =

1, 12 0.4


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FIG. 6.36. Degenerate parabolic problemsolution at time T= 0.5 on a 60 60 grid.

or (u)= 0 (hyperbolic problem). The initial data are equal to1 and 1 inside two circles ofradius 0.4 centered at (0.5, 0.5) and (

0.5,

0.5), respectively, and zero elsewhere inside the

square [1.5, 1.5] [1.5, 1.5]. As in the one-dimensional examples, our scheme performswell in both hyperbolic and hyperbolicparabolic cases even with relatively small number

of grid points.

EXAMPLE 12 (Two-Dimensional BuckleyLeverett Equation). Finally, we solve the

two-dimensional convectiondiffusion equation

ut+ f(u)x + g(u)y = (ux x + uyy ), (6.19)

with = 0.01, the flux function of the form

f(u) = u2

(u2 + (1 u)2) ,(6.20)

g(u) = f(u)(1 5(1 u)2),and the initial data

u(x , y, 0) = 1, x2

+y2 < 0.5,

0, otherwise. (6.21)

Note that the above model includes gravitational effects in the y-direction.

FIG. 6.37. Problem (6.19)(6.21)solution at time T= 0.5 on a 200 200 grid.


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FIG. 6.38. Problem (6.19)(6.21)solution at time T= 0.5 on a 100 100 grid.

The solution, computed in the domain [1.5, 1.5] [1.5, 1.5] by the SD2 scheme, isshown in Figs. 6.37 and 6.38. Our scheme also provides a highly satisfactory approximation

to this model with a nonlinear, degenerate diffusion.

ACKNOWLEDGMENT

The authors thank Dr. A. Medovikov (Institute of Numerical Mathematics, Russian Academy of Sciences,

Russia) for a number of useful discussions and for providing his ODE solver DUMKA3. The work of A. Kurganov

was supported in part by the NSF Group Infrastructure Grant. The work of E. Tadmor was supported in part by

NSF Grant DMS97-06827 and ONR Grant N00014-91-J-1076.

REFERENCES

1. A. M. Anile, V. Romano, and G. Russo, Extended hydrodynamical model of carrier transport in semiconduc-tors, SIAM J. Appl. Math., in press.

2. P. Arminjon, D. Stanescu, and M.-C. Viallon, A two-dimensional finite volume extension of the LaxFriedrichs

and NessyahuTadmor schemes for compressible flow, in Proc. 6th Int. Symp. on CFD, Lake Tahoe, 1995,

M. Hafez and K. Oshima, editors, Vol. IV, pp. 714.

3. P. Arminjon and M.-C. Viallon, Generalisation du schema de NessyahuTadmor pour une equation hyper-

bolique a deux dimensions despace, C. R. Acad. Sci. Paris Ser. I320, 85 (1995).

4. F. Bianco, G. Puppo, and G. Russo, High order central schemes for hyperbolic systems of conservation laws,

SIAM J. Sci. Comput. 21, 294 (1999).5. P. Colella, private communication.

6. H. K. Dahle, Adaptive Characteristic Operator Splitting Techniques for Convection-Dominated Diffusion

Problems in One and Two Space Dimensions, Ph.D. thesis, University of Bergen, Norway, 1988.

7. B. Engquist and O. Runborg, Multi-phase computations in geometrical optics,J. Comput. Appl. Math. 74,

175 (1996).


41/42


8. S. Evje, K. H. Karlsen, K.-A. Lie, and N. H. Risebro, Front tracking and operator splitting for nonlinear

degenerate convectiondiffusion equations, Institut Mittag-Leffler Report, Stockholm, Sweden, 1997.

9. A. C. Fowler, Glaciers and ice sheets, inThe Mathematics of Models for Climatology and Environment, edited

by J. I. Diaz, NATO ASI Series, Vol. 48 (Springer-Verlag, Berlin/New York, 1996), p. 302.

10. K. O. Friedrichs, Symmetric hyperbolic linear differential equations,Comm. Pure Appl. Math. 7, 345 (1954).

11. S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the

equations of fluid dynamics, Mat. Sb. 47, 271 (1959).

12. J. Goodman, A. Kurganov, and P. Rosenau, Breakdown of Burgers-type equations with saturating dissipation

fluxes, Nonlinearity 12, 247 (1999).

13. A. Harten, High resolution schemes for hyperbolic conservation laws,J. Comput. Phys. 49, 357 (1983).

14. A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy, Uniformly high order accurate essentially non-

oscillatory schemes, III, J. Comput. Phys. 71, 231 (1987).

15. A. Harten, P. D. Lax, and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic

conservation laws, SIAM Rev. 25, 35 (1983).

16. A. Jameson, Internat. J. Comput. Fluid Dynamics 4, 171 (1995); 5, 1 (1995).

17. G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher, and E. Tadmor, High-resolution non-oscillatory central schemes

with non-staggered grids for hyperbolic conservation laws,SIAM J. Numer. Anal. 35, 2147 (1998).

18. G.-S. Jiang and E. Tadmor, Non-oscillatory central schemes for multidimensional hyperbolic conservation

laws, SIAM J. Sci. Comput. 19, 1892 (1998).

19. S. Jin and Z. Xin, The relaxation schemes for hyperbolic systems of conservation laws in arbitrary space

dimensions, CPAM48, 235 (1995).

20. K. H. Karlsen, K. Brusdal, H. K. Dahle, S. Evje, and K.-A. Lie, The corrected operator splitting approach

applied to a nonlinear advectiondiffusion problem,Comput. Methods Appl. Mech. Eng. 167, 239 (1998).

21. K. H. Karlsen and K.-A. Lie, An unconditionally stable splitting for a class of nonlinear parabolic equations,

IMA J. Numer. Anal. 19, 609 (1999).

22. K. H. Karlsen and N. H. Risebro, An operator splitting method for nonlinear convectiondiffusion equations,

Numer. Math. 77, 365 (1997).

23. K. H. Karlsen and N. H. Risebro, Corrected operator splitting for nonlinear parabolic equations,SIAM J.

Numer. Anal. 37, 980 (2000).

24. R. Kupferman, Simulation of viscoelastic fluids: CouetteTaylor flow,J. Comput. Phys., to appear.

25. R. Kupferman, A numerical study of the axisymmetric CouetteTaylor problem using a fast high-resolution

second-order central scheme, SIAM J. Sci. Comput. 20, 858 (1998).

26. R. Kupferman and E. Tadmor, A fast high-resolution second-order central scheme for incompressible flows,

Proc. Nat. Acad. Sci. 94, 4848 (1997).

27. A. Kurganov, Conservation Laws: Stability of Numerical Approximations and Nonlinear Regularization,

Ph.D. thesis, Tel-Aviv University, Israel, 1997.

28. A. Kurganov and P. Rosenau, Effects of a saturating dissipation in Burgers-type equations,Comm. Pure Appl.Math. 50, 753 (1997).

29. P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation,Comm. Pure

Appl. Math. 7, 159 (1954).

30. B. van Leer, Towards the ultimate conservative difference scheme. III. Upstream-centered finite-difference

schemes for ideal compressible flow, J. Comput. Phys. 23, 263 (1977).

31. B. van Leer, Towards the ultimate conservative difference scheme. V. A second order sequel to Godunovs

method, J. Comput. Phys. 32, 101 (1979).

32. D. Levy, G. Puppo, and G. Russo, Central WENO schemes for hyperbolic systems of conservation laws,Math. Model. Numer. Anal. 33, 547 (1999).

33. D. Levy and E. Tadmor, Non-oscillatory central schemes for the incompressible 2-D Euler equations,Math.

Res. Lett. 4, 1 (1997).

34. X.-D. Liu and P. D. Lax, Positive schemes for solving multidimensional systems of hyperbolic conservation

laws, Comput. Fluid Dynamics J. 5, 1 (1996).


42/42


35. X.-D. Liu and S. Osher, Convex ENO high order multi-dimensional schemes without field by field decompo-

sition or staggered grids, J. Comput. Phys. 142, 304 (1998).

36. X.-D. Liu and E. Tadmor, Third order nonoscillatory central scheme for hyperbolic conservation laws,Numer.

Math. 79, 397 (1998).

37. A. A. Medovikov, High order explicit methods for parabolic equations,BIT38, 372 (1998).

38. H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws,J. Comput.

Phys. 87, 408 (1990).

39. H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for non-linear scalar conservation

laws, SIAM J. Numer. Anal. 29, 1505 (1992).

40. S. Osher, Convergence of generalized MUSCL schemes, SINUM22, 947 (1984).

41. S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws,Math.

Comput. 50, 19 (1988).

42. P. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes,J. Comput. Phys. 43, 357

(1981).

43. V. Romano and G. Russo, Numerical solution for hydrodynamical models of semiconductors, M3AS, preprint.

44. C.-W. Shu, Total-variation-diminishing time discretizations,SISSC6, 1073 (1988).

45. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes,

J. Comput. Phys. 77, 439 (1988).

46. G. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws,

J. Comput. Phys. 22, 1 (1978).

47. E. Tadmor, Convenient total variation diminishing conditions for nonlinear difference schemes,SIAM J.

Numer. Anal. 25, 1002 (1988).

new high-resolution central schemes for nonlinear conservation laws and convection diffusion...

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